Providing you are confident she is safe with him (which you clearly are, else you wouldn't be sending her in the first place) then she should stay.
Children, just like adults, have to be places they don't want to be sometimes. If she wanted to be picked up from school, you wouldn't do it, just because she preferred being at home.
It's totally natural that, as she isn't used to over nights at her dad's, it will take a bit of time for her to adjust to it. But it's a good thing for her to spend time with him, and develop that tiny bit of independence from you.
I understand that it must feel hurtful to him at the moment, that feeling that his daughter doesn't want to spend time with him. But she clearly DOES love him (she misses him in the week) so really you all just have to power through this tricky bit to get to a point where you are all more settled and happy, and that she is used to staying with him.
I split with my dd's dad when she was about 5, and I was the one to move out of the family home. We have always shared custody, but initially she really did not want to stay with me in my new house. We had some truly horrific scenes and I felt utterly broken by it.
She now speaks really positively about the experience she had growing up in two different homes and in what are essentially two very different lifestyles - yes she is a remarkable person, but she essentially has managed the transition smoothly (with our support and our truly EXCELLENT co-parenting - we were a terrible couple but have done a great job raising her separately!) , and is equally at home now spending the summer travelling the country living out of a van with me, or socialising at a restaurant in Soho with her dad's extremely wealthy friends (we really are VERY different people! Tho both good parents I believe, each bringing something very different to the table. And have both always put her at the centre of our lives in some ways yes, though not to the exclusion of HAVING our own lives, IYSWIM).
Try and think of some practical things that would make her feel more happy at exes - are there certain toys she would like to have there, foods or activities he doesn't offer currently, etc? I am not saying he should totally revamp his house to suit her tastes, but maybe there are things he could do to make it feel more like a home to her?
And then when all that is in place, next time she asks to come home, it does just have to be "sorry, darling, but no - this is your time with dad, you will see mom soon".
It might also be worth gently explaining to her that it really hurts his feelings when she says she doesn't want to spend time with him? It doesn't hurt children to know that other people have feelings that matter too, and that they have the power to hurt those with their actions ....
Good luck! FWIW you sound like a great mom, and she is lucky to have you!